Partitioning complete graphs by heterochromatic trees

TL;DR

This paper determines the minimum number of heterochromatic trees needed to cover the vertices of an edge-colored complete graph, advancing understanding of heterochromatic tree partitions in graph theory.

Contribution

It establishes the heterochromatic tree partition number for complete graphs under any edge coloring with r colors, providing a precise value.

Findings

Exact heterochromatic tree partition number for complete graphs.

Generalization to any number of edge colors r.

Improved bounds for specific cases.

Abstract

A {\it heterochromatic tree} is an edge-colored tree in which any two edges have different colors. The {\it heterochromatic tree partition number} of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum positive integer $p$ such that whenever the edges of the graph $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $p$ vertex-disjoint heterochromatic trees. In this paper we determine the heterochromatic tree partition number of an $r$-edge-colored complete graph.